Let . Prove that:
Proof
We have
Let then we have
We have
From Holder’s inequality we have
But we have
From AM-GM inequality we have
Therefore the proof is completed. Equality hold if
Let . Prove that:
Proof
We have
Let then we have
We have
From Holder’s inequality we have
But we have
From AM-GM inequality we have
Therefore the proof is completed. Equality hold if
Let . Prove that:
Proof
Let then we need to prove that:
We have
Because then from inequality we get:
From inequality we have
From inequality we have
Therefore the proof is completed. Equality holds for